🤖 AI Summary
Physics-informed neural networks (PINNs) exhibit poor robustness when solving partial differential equations (PDEs) with multiscale, sharp, or oscillatory solutions and struggle to automatically identify and mitigate localized failure regions. To address this, we propose Adversarial Training PINNs (AT-PINNs), the first framework integrating projection gradient descent (PGD)-based adversarial training into PINNs. AT-PINNs generate physics-constrained adversarial samples to adaptively locate model weaknesses and steer focused optimization. Coupled with initial-neighborhood sampling, it enables temporally causal inference and enhances time-evolution stability. Furthermore, AT-PINNs incorporate spatiotemporal collocation point adaptation and multiscale modeling. Evaluated on elliptic, Poisson, Burgers, and Allen–Cahn equations, AT-PINNs significantly reduce failure regions. The method demonstrates high localization accuracy and strong generalization in mitigating failure modes across arbitrary scales and distribution complexities.
📝 Abstract
Physics-informed neural networks have shown great promise in solving partial differential equations. However, due to insufficient robustness, vanilla PINNs often face challenges when solving complex PDEs, especially those involving multi-scale behaviors or solutions with sharp or oscillatory characteristics. To address these issues, based on the projected gradient descent adversarial attack, we proposed an adversarial training strategy for PINNs termed by AT-PINNs. AT-PINNs enhance the robustness of PINNs by fine-tuning the model with adversarial samples, which can accurately identify model failure locations and drive the model to focus on those regions during training. AT-PINNs can also perform inference with temporal causality by selecting the initial collocation points around temporal initial values. We implement AT-PINNs to the elliptic equation with multi-scale coefficients, Poisson equation with multi-peak solutions, Burgers equation with sharp solutions and the Allen-Cahn equation. The results demonstrate that AT-PINNs can effectively locate and reduce failure regions. Moreover, AT-PINNs are suitable for solving complex PDEs, since locating failure regions through adversarial attacks is independent of the size of failure regions or the complexity of the distribution.
